Generally speaking, piracy during the age of discovery was limited by a) the seagoing qualities of the ships of the age, and by the fact that piracy doesn't pay unless you have b) lots of vulnerable merchant shipping and coastal towns with c) little protection from local overlords.

As for reason a) goes, it was not at all easy to intercept a sailing ship anywhere but near its point of departure or destination. Sea is vast and visibility is limited even at the best of times, and even if one were able to spot another vessel, the chances were that the both vessels' sailing qualities were close enough so that a chase was unprofitable. Hence, most attacks happened against ships found lying to in or close to a harbor, not on the high seas or specifically not in unexplored regions of the world.

As for b) and c) go, piracy is practically nonexistent where the locals are well protected or where there are no pickings to be made. Sure enough, in time of Columbus's first voyage, the Caniba indians in the Caribbean for example were claimed to raid neighboring islands for loot and captives, which they then ate (incidentally, hence the word "cannibal"). Columbus however had no contact with their inhabitants.

If we look at the early explorers, the real threat of piracy was in the Orient. There were lots of maritime traffic and lack of effective law enforcement (so to speak). However, the European explorers generally speaking had far superior ships and to my knowledge there are no accounts of them being beset by actual sea-going pirates. One example illustrates this superiority fairly well: the Portuguese expedition immediately following Vasco da Gama's famous voyage, this one led by Pedro Alvares Cabral, became pirates when he sent one of his smaller caravels to seize a ship in Calicut harbor. The ship belonged to the rival of the ruler of Calicut and was given to him as a present. The ruler, or zamorin, was reported to have "marveled greatly that so small a caravel and with so few people could take so large a ship in which were 300 men at arms." (The source here is "The Explorers" by Richard Humble et al., 1979, p. 105).